Catholic and independent schools will receive an extra $4.6bn over the next decade in a peace deal with the two non-government sectors after the Catholic sector bitterly opposed the Coalition’s 2017 schools funding changes.

The prime minister, Scott Morrison, and the education minister, Dan Tehan, announced the funding fix on Thursday, after months of backroom negotiations started by former minister Simon Birmingham before Malcolm Turnbull was deposed.

The deal introduces a new direct measure of parents’ income to determine schools funding, as recommended by the Chaney review, and gives “overfunded” schools an extra two years to transition down to 80% of the schools resource standard.

It also creates a $1.2bn fund for ill-defined priorities to improve “school choice and affordability”, which Tehan described as “sector-blind” despite the fact only independent and Catholic schools can apply. Guardian Australia understands the fund was not on the table until the last month, after the Liberal leadership and education portfolio change.

The agreement was welcomed by both the National Catholic Education Commission and the Independent Schools Council of Australia but blasted by Labor and the Australian Education Union, which cited the fact it did nothing for public schools.

But the former education minister in New South Wales, National Adrian Piccoli, was its fiercest critic, labelling the deal “pathetic” and accusing the government of “throwing money at the powerful and well-connected”.

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Morrison told reporters in Canberra the package demonstrated the government “believes in choice in education”. He defended helping the Catholic and independent sectors by arguing the federal government “has always been the principal funder of non-government schools”.

The deal consists of three measures:

$170.8m in interim funding in 2019 including a guarantee that independent schools get at least 3% funding growth

$3.2bn from 2020 to 2029 by replacing the socio-economic status score of communities’ capacity to contribute to schools with a new direct measure of parents’ income

$1.2bn from 2020 to 2029 for the Choice and Affordability Fund to be allocated for government priorities including diversity, regional, rural and remote education

Of the total $4.6bn package, about $1.1bn will be spent in the next four years.

Morrison did not commit to spending cuts to pay for the package, instead citing increased revenue due to improved business performance and promising the extra spending will be “reconciled” in December’s mid-year economic and fiscal outlook.

The Australian Education Union federal president Correna Haythorpe said the policy was “a cynical attempt by the Morrison government to buy votes at the next election at the expense of students in our public schools”.

“Mr Morrison may think he has settled the funding wars but he is wrong,” Haythorpe said. “We will escalate our campaign in 18 target seats, ensuring that parents across Australia know that it is the Morrison government which has abandoned public school students.”

Labor’s education spokeswoman, Tanya Plibersek, said the deal “looks desperate” and amounted to an admission the Coalition had cut billions from schools but did nothing to rectify $14bn of cuts to public schools.

She said it was “a completely inadequate response to the $17bn cut from schools over the next decade”.

Morrison defended the fairness of the package by arguing that government schools still received “record” funding levels and that state governments were the primary funders of government schools.

The National Catholic Education Commission acting executive director, Ray Collins, commended Tehan “for recognising that the 2017 changes had jeopardised the future of low-fee, low-expenditure schools in areas where they’ve served families for generations”.

The NCEC believes the SES model of funding discriminates against low-fee Catholic schools because they attract a higher proportion of students from poorer families and therefore need more funding support.

Despite “fully supporting” the changes, Collins said that “fundamental to our support … is the minister’s agreement to review the new arrangements to ensure they continue to support the government’s policy objectives, including parent choice”.

The Independent Schools Council of Australia said the new funding deal “creates the foundation for a fair and reasonable resolution of the current funding issues”.

“Many independent schools could, over the long term, face significant changes in government funding, and for this reason ISCA supports the announcement of transitional measures that will help form the basis for a long-term funding model that is sector-blind and focused on the needs of students,” it said.

Under the new model, individual schools will be able to choose whether to use the new direct income measure in 2020, 2021 or 2022, when all schools will use the new system.

The new model preserves Gonski 2.0’s central design – that non-government schools will receive 80% of the school resource standard from the federal government – but gives overfunded schools until 2029 (out from 2027) to transition down to that level. Non-government schools below the 80% mark will reach it by 2023.